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Ultrafilters, monotone functions and pseudocompactness

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Abstract.

In this article we, given a free ultrafilter p on ω, consider the following classes of ultrafilters:

(1) T(p) - the set of ultrafilters Rudin-Keisler equivalent to p,

(2) S(p)={q ∈ ω*:∃ f ∈ ωω, strictly increasing, such that q=fβ(p)},

(3) I(p) - the set of strong Rudin-Blass predecessors of p,

(4) R(p) - the set of ultrafilters equivalent to p in the strong Rudin-Blass order,

(5) P RB (p) - the set of Rudin-Blass predecessors of p, and

(6) P RK (p) - the set of Rudin-Keisler predecessors of p,

and analyze relationships between them. We introduce the semi-P-points as those ultrafilters p ∈ ω* for which P RB (p)=P RK (p), and investigate their relations with P-points, weak-P-points and Q-points. In particular, we prove that for every semi-P-point p its α-th left power αp is a semi-P-point, and we prove that non-semi-P-points exist in ZFC. Further, we define an order ⊴ in T(p) by r⊴q if and only if r ∈ S(q). We prove that (S(p),⊴) is always downwards directed, (R(p),⊴) is always downwards and upwards directed, and (T(p),⊴) is linear if and only if p is selective.

We also characterize rapid ultrafilters as those ultrafilters p ∈ ω* for which R(p)∖S(p) is a dense subset of ω*.

A space X is M-pseudocompact (for ) if for every sequence (U n ) n < ω of disjoint open subsets of X, there are q ∈ M and x ∈ X such that x=q-lim (U n ); that is, for every neighborhood V of x. The P RK (p)-pseudocompact spaces were studied in [ST].

In this article we analyze M-pseudocompactness when M is one of the classes S(p), R(p), T(p), I(p), P RB (p) and P RK (p). We prove that every Frolik space is S(p)-pseudocompact for every p ∈ ω*, and determine when a subspace with is M-pseudocompact.

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References

  1. Bartoszyński, T., Judahm, H: Set Theory, On the Structure of the Real Line. A. K. Peters, 1995

  2. Bernstein, A.R.: A new kind of compactness for topological spaces. Fund. Math. 66, 185–193 (1970)

    MATH  Google Scholar 

  3. Blass, A.R.: Kleene degrees of ultrafilters. In: Recursion Theory Week (Oberwolfach, 1984), Lecture Notes in Math., vol. 1141, Springer, Berlin-New York, 1985, pp. 29–48

  4. Booth, D.D.: Ultrafilters on a countable set. Ann. Math. Logic. 2, 1–24 (1970)

    Article  Google Scholar 

  5. Frolík, Z.: The topological product of two pseudocompact spaces. Czechoslovak Math. J. 85, 339–349 (1960)

    Google Scholar 

  6. Frolík, Z.: Sums of ultrafilters. Bull. Amer. Math. Soc. 73, 187–191 (1967)

    Google Scholar 

  7. García-Ferreira, S.: Some generalizations of pseudocompactness. Ann. New York Acad. Sci. 728, 22–31 (1994)

    Google Scholar 

  8. García-Ferreira, S., Sanchis, M.: On C-compact subsets. Houston. J. Math. 23, 65–86 (1997)

    Google Scholar 

  9. García-Ferreira, S., Tamariz-Mascarúa, A.: On p-sequential p-compact spaces. Comm. Math. Univ. Carolinae 34, 347–356 (1993)

    Google Scholar 

  10. Ginsburg, J., Saks, V.: Some applications of ultrafilters in topology. Pacific J. Math. 57, 403–418 (1975)

    MATH  Google Scholar 

  11. Katětov, M.: Products of filters. Comment. Math. Univ. Carolin. 9, 173–189 (1968)

    Google Scholar 

  12. Kunen, K.: Weak P-points in N*. Coll. Math. Soc. János Bolyai 23, Topology, Budapest (Hungary), 1978, pp. 741–749

  13. Laflamme, C., Zhu, J.: The Rudin-Blass ordering on ultrafilters. J. Symb. Logic. 63, 584–592 (1998)

    MathSciNet  MATH  Google Scholar 

  14. van Mill, J.: An introduction to βω. In: Handbook of Set Theoretic Topology, K. Kunen, J. Vaughan (eds.), North-Holland, Amsterdam, 1984

  15. Miller, A.: There are no Q-points in Laver’s model for the Borel conjecture. Proc. Amer. Math. Soc. 78, 103–106 (1980)

    MathSciNet  MATH  Google Scholar 

  16. Noble, N.N.: Countably compact and pseudocompact products. Czechoslovak Math. J. 19, 390–397 (1969)

    MATH  Google Scholar 

  17. Sanchis, M., Tamariz-Mascarúa, A.: On quasi-p-bounded subsets. Colloquium Math. 80, 175–189 (1999)

    MathSciNet  MATH  Google Scholar 

  18. Sanchis, M., Tamariz-Mascarúa, A.: A note on p-bounded and quasi-p-bounded subsets. Houston. J. Math. 28, 511–527 (2002)

    Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Xangari, 58089, Morelia Michoacan, México

    M. Hrušák

  2. Departament de Matemàtiques, Universitat Jaume I, Campus de Riu Sec s/n, Castelló, Spain

    M. Sanchis

  3. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, 04510, México

    Á. Tamariz-Mascarúa

Authors
  1. M. Hrušák
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  2. M. Sanchis
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  3. Á. Tamariz-Mascarúa
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Correspondence to M. Hrušák.

Additional information

The first author’s research was partially supported by a grant GAČR 201/00/1466

Mathematics Subject Classification (2000): 54D80, 03E05, 54A20, 54D20

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Hrušák, M., Sanchis, M. & Tamariz-Mascarúa, Á. Ultrafilters, monotone functions and pseudocompactness. Arch. Math. Logic 44, 131–157 (2005). https://doi.org/10.1007/s00153-004-0246-y

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  • DOI: https://doi.org/10.1007/s00153-004-0246-y

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